Testing the Dominance of Biometric Functions
 
M. Kayida, I.A. Al-Waselaand I.A. Ahmadb
 
aDepartment of Statistics and Operations Research, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Kingdom of Saudi Arabia
bDepartment of Statistics, Oklahoma State University, Stillwater, OK 74078, USA
 
The hazard rate (HR) and mean residual lifetime are two of the most practical and best-known functions in biometry, reliability, statistics and life testing. Recently, the reversed HR function is found to have interesting properties useful in additional areas such as censored data and forensic science. For these three biometric functions, we propose testing methods that they take on a known functional form against that they dominate or are dominated by this known form. This goodness-of-fit-type testing is wider in applications and more interesting than the long-standing testing procedures for exponentially against the monotonicity of these functions or even the change point problems. This is so since we can test against any choice of the survival distribution and not just exponentially. For this general testing, we present easy to implement tests and generalize them into classes of statistics that could lead to more powerful and efficient testing.
 
Keywords:hazard rate; mean residual lifetime; reversed hazard rate; test statistics; asymptotic normality; efficiency; U-statistics
 
Testing Behavior of the Reversed Hazard Rate
M. Kayid1 and H. Al-nachawati
King Saud University, College of Science, Dept. of Statistics & Operations Research,
P.O. Box 2455, Riyadh 11451.
 
Ibrahim Ahmad
Department of Statistics, Oklahoma State University,
Stillwater, OK 74078, USA
 
Abstract
The concept of reversed hazard rate of a random life is defined as the ratio between the life probability density to its distribution function. This concept plays a role in analyzing censored data and is applicable in such areas as Forensic Sciences. In this investigation, we address the question of testing the reversed hazard rate where the null is that the reversed hazard rate is an assigned function while the alternative is that it is decreasing but not equal to the null function. Two approaches are discussed: one is based on the empirical distribution function while the other is based on the celebrated kernel methods. The limiting distributions of the test statistics are given and its asymptotic Pittman efficiencies are evaluated for well-known alternatives when thenull distribution is exponential. Some other related problems are also addressed.
 
Keywords: Empirical distribution, test statistic, concave (convex) functions, kernel, bandwidth, asymptotic normality, Pittman efficacy.

doi:10.4236/iim.2010.211072 Published Online November 2010 (http://www.SciRP.org/journal/iim)
Copyright © 2010 SciRes. IIM
Record Values from the Inverse Weibull Lifetime Model:
Different Methods of Estimation
Khalaf S. Sultan
Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh, Saudi Arabia
E-mail: ksultan@ksu.edu.sa
Received September 21, 2010; revised October 13, 2010; accepted November 15, 2010
Abstract
 
In this paper, we use the lower record values from the inverse Weibull distribution (IWD) to develop and discuss different methods of estimation in two different cases, 1) when the shape parameter is known and 2) when both of the shape and scale parameters are unknown. First, we derive the best linear unbiased estimate (BLUE) of the scale parameter of the IWD. To compare the different methods of estimation, we present the results of Sultan (2007) for calculating the best linear unbiased estimates (BLUEs) of the location and scale parameters of IWD. Second, we derive the maximum likelihood estimates (MLEs) of the location and scale parameters. Further, we discuss some properties of the MLEs of the location and scale parameters. To compare the different estimates we calculate the relative efficiency between the obtained estimates. Finally, we propose some numerical illustrations by using Monte Carlo simulations and apply the findings of the paper to some simulated data.
Keywords: Scale Parameter, Location Parameter, Best Linear Unbiased Estimates (BLUEs), Maximum
Likelihood Estimates, Relative Efficiency and Monte Carlo Simulations
 
Advanced Nonlinear Studies 10 (2010), 771{788
The Wente Problem Associated to the Modi¯ed
Helmholtz Operator on Weighted Sobolev Spaces
Ines Ben Omrane
D¶epartement de Math¶ematiques
Facult¶e des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia
e-mail: benomraneines@gmail.com
Mohamed Jleli¤
Department of Mathematics
King Saud University, Saudi Arabia
e-mail: jleli@ksu.edu.sa
Bessem Samet
D¶epartement de Math¶ematiques
Ecole Sup¶erieure des Sciences et Techniques de Tunis, 5 Avenue Taha Hussein, 1008, Tunisia
e-mail: bessem.samet@gmail.com
Communicated by Abbas Bahri
Abstract
In this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modi¯ed Helmholtz operator ¡¢ + ®I, where ® is apositive constant.
 
Some Results on the Excess Wealth Order with Applications in Reliability Theory
 
M. Kayidand S. Al-Dokar
Department of Statistics and Operations Research
KingSaud University, Riyadh 11451, Saudi Arabia
 
Abstract: In this paper, we establish some characterizations of the excess wealth order. As a consequence of our results, we provide some applications in reliability theory and characterize some well known classes of life distributions.
 
Key Words : Stochastic orders, proportional hazard (reversed hazard), model, DMRL, IMIT, NBUT.

On the Convolution Order with Reliability Applications
 
A. Alzaid and M. Kayid
 
King Saud University, College of Science
Dept. of Statistics and Operation Research
P.O. Box 2455, Riyadh 11451, Saudi Arabia
Abstract
In this paper, we study further the convolution order and investigate its reliability properties. We apply a simple useful property of this order to families of non-negative random variables that have the generalized semi group property. Some characterizations and applications in reliability theory are described.
 
Keywords: Convolution order, Laplace transform order, Laplace transform ratio order, shock models, generalized semi-group property, self-decompossible distributions
 
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 1, 17 - 29
On Testing Statistics of Renewal New Better
than Renewal Used Class of Life Distributions
I. Elbatal
King Saud University, College of Science
Department of Statistics and Operation Research
P.O. Box 2455, Riyadh 11451, Kingdom of Saudi Arabia
ielbatal@yahoo.com, ielbatal@ksu.edu.sa
Abstract
A moment inequality for the class of renewal new is better than renewal used in (RNBRU) of ageing distributions is derived. This class is defined based on comparing the residual equilibrium life at a certain age and its equilibrium (stationary) life . This inequality demonstrate
that if the mean life is finite,then all higher order moments exist. A new test statistics for testing exponentiality against RNBRU is investigating based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman’s asymptotic efficiency of the test and
critical values of the proposed statistic are calculated. It is shown that ,the proposed statistic has a high asymptotic relative efficiency with respect to tests of other classes for commonly used alternatives. The set of real data is used as a practical application of the proposed test in the medical science.
Keywords: Life distributions, RNBRU, Moments inequalities, Testing
Exponentiality, Asymptotic normality, Efficiency
  
Applied Mathematical Sciences, Vol. 3, 2009, no. 11, 541 - 550
Parameters Estimation of the Modified
Weibull Distribution
Mazen Zaindin
Department of Statistics and Operations Research
P.O. Box 2455, Riyadh 11451, Saudi Arabia
mazenzaindin@yahoo.com
Ammar M. Sarhan
Department of Mathematics and Statistics
Faculty of Science, Dalhousie University
Halifax NS B3H 3J5, Canada
asarhan@mathstat.dal.ca, asarhan0@yahoo.com
Abstract
Recently, Sarhan and Zaindin (2008) introduced a generalization of the Weibull distribution and named it as modified Weibull distribution. In this paper, we deal with the problem of estimating the parameters of this distribution based on Type II censored data. The maximum likelihood and least square techniques are used. For illustrative purpose, the results obtained are applied on sets of real data. Also, simulation is used to study the properties of the estimators derived.
Keywords: Maximum likelihood, least square, linear failure rate, Rayleigh,
exponential distribution
 
Optimal control of an inventory system with ameliorating and
deteriorating items
Lot¯ Tadj, Ammar M. Sarhan, and Awad El-Gohary
Abstract
This paper is concerned with the development of a production inventory model for both amelio- rating and deteriorating items. Given an inventory goal level and a production goal rate set by the production facility, and given penalties for the inventory level and for the production rate to deviate from their respective targets, a system total cost objective function is derived. We seek the optimal production rate, that is the production rate that minimizes this performance measure, while satisfying the system dynamics. The resulting problem is an optimal control problem with mixed inequality constraints, in which the inventory level is the state variable and the production rate is the control variable. The necessary optimality conditions are derived using Pontryagin maximum principle. This paper generalizes some of the models available in the literature.
M.S.C. 2000: 49J15, 90B30.
Key words: Inventory systems, production planning, deteriorating items, ameliorating items, tar-
get, optimal control.
 
Applied Mathematical Sciences, Vol. 3, 2009, no. 12, 591 - 604
The Chaos and Control of Food Chain Model
Using Nonlinear Feedback1
A. Al-Khedhairi
Department of Statistics and O. R., College of Science
KingSaud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
akhediri@ksu.edu.sa
Abstract
This paper studies the chaos and control of a continuous time food chain model which contains one prey, one predator and super-predator. We show that this system can be asymptotically stabilized using a nonlinear feedback control inputs. The necessary feedback control law for asymptotic stability of this system is obtained. The system appears to exhibit a chaotic behavior for a range of parametric values. The range of the system parameters for which the subsystems converge to limit cycles is determined. Numerical examples and analysis of the results are presented.
Keywords: Food chain, Limit cycles, Chaotic behavior, chaos control,
Asymptotic stability, Liapunov function

Quality and Reliability Engineering International
Qual. Reliab. Engng. Int. 2012, 28 761–766
Research Article1
(wileyonlinelibrary.com) DOI: 10.1002/qre.1272
A Goodness of Fit Approach to the Class of Life Distributions with Unknown Age
Ali A. Ismail* and S. E. Abu-Youssef
* King Saud University, College of Science, Department of Statistics and OR, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
 
Abstract:Based on the goodness of fit approach, a new test is presented for testing exponentiality against the unknown age (used better than aged in convex ordering (UBAC)) Class of life distributions. The percentiles of this test are tabulated for sample sizes n=5(5)40. It is shown that the proposed test is simple, has high relative efficiency for some commonly used alternatives and enjoys a good power. An example in medical science is considered as a practical application of the proposed test.
 
Keywords:Reliability; goodness of fit approach; unknown aging class; hypothesis testing; asymptotic normality; efficiency; power.